Finite Group Databasebeta

← #196847

Subgroups — #196847

Order 512.

1566 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35178C211yesmaximal, characteristic
2C2×#2351C211yesmaximal, characteristic, complemented
3#21818C211yesmaximal, characteristic, complemented
4C2×#1517C211yesmaximal, characteristic
5C2×#1502C211yesmaximal, characteristic
6#9156C214yesmaximal, complemented
7#9148C214yesmaximal, complemented
8#8026C211yesmaximal, characteristic, complemented
9#8014C211yesmaximal, characteristic, complemented
10C2×#536—21no—
11C2×#532C2211yescharacteristic
12C22×C4×Q8—21no—
13C22×C4×Q8C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C2×#431C2211yescharacteristic
16#2854C2214yes—
17#2618—22no—
18#2548C412yescomplemented
19#2546C412yescomplemented
20#2545—22no—
21#2538C414yescomplemented
22#2536—22no—
23#2351—24no—
24#2351—24no—
25#2311—21no—
26C2×#389—21no—
27C2×#389—22no—
28#2285C2211yescharacteristic
29C2×#382C2212yes—
30C2×#382C2211yescharacteristic
31C2×C4×C4⋊C4C2212yes—
32C2×C4×C4⋊C4C2211yescharacteristic
33#1984—22no—
34#1984—22no—
35#1817C2214yes—
36#1517—24no—
37#1502—24no—
38C2×#327C2212yes—
39C2×#327C2212yes—
40C2×#327C2214yes—
41#1294C2212yes—
42#1294C2212yes—
43#1294C2214yes—
44C22×#41—21no—
45C23×Q8D411yescharacteristic
46#536—44no—
47#536—44no—
48#532—22no—
49#532—22no—
50#532—22no—
51#532—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×#127C2×C411yescharacteristic
55C2×#124—22no—
56C2×#124—21no—
57C2×#123—24no—
58C2×C22⋊Q8—24no—
59C2×#121—21no—
60C2×#121C2312yes—
61C2×C4×Q8—28no—
62C2×C4×Q8—22no—
63C2×C4×Q8—22no—
64C2×C4×Q8—22no—
65C2×C4×Q8—22no—
66C2×C4×Q8—44no—
67C2×C4×Q8—44no—
68C2×C4×Q8C2×C412yes—
69C2×C4×Q8C2×C412yes—
70C2×C4×Q8D412yes—
71C2×C4×Q8D412yes—
72C2×C4×D4—21no—
73C2×C4×D4C2×C412yes—
74C2×#117—21no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—22no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4D411yescharacteristic
80C22×C4⋊C4D412yes—
81C22×C42—22no—
82C22×C42—21no—
83C22×C42C2312yes—
84C22×C42C2311yescharacteristic
85#431—22no—
86#431—22no—
87#431—22no—
88#431—22no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C22⋊C8C2312yes—
92C2×C22⋊C8C2312yes—
93C2×C4×C8—21no—
94C2×C4×C8—21no—
95#398—24no—
96#396—28no—
97#394—24no—
98#391—22no—
99#390—22no—
100#390C2×C418yes—
101#389—28no—
102#389—216no—
103#389C2×C418yes—
104#388—28no—
105#386—24no—
106#384C2×C412yes—
107#382—22no—
108#382—21no—
109#382—24no—
110#382—24no—
111#382D414yes—
112#380—21no—
113#380—24no—
114#380C2312yes—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—24no—
118C4×C4⋊C4—24no—
119C4×C4⋊C4C2×C412yes—
120C4×C4⋊C4D414yes—
121C2×#95—22no—
122C2×#95—21no—
123C2×#95C2311yescharacteristic
124C2×#95C2312yes—
125C2×#95C2311yescharacteristic
126#336—28no—
127#327—24no—
128#327—24no—
129#327—28no—
130C2×#41—42no—
131C2×#41—42no—
132C2×#41—42no—
133C2×#41—42no—
134C2×#41—42no—
135C2×#41—42no—
136C22×Q8—21no—
137C22×Q8—22no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8C22⋊C412yes—
142C22×Q8C22⋊C412yes—
143C22×Q8D814yes—
144C22×Q8SD1614yes—
145C22×D4—22no—
146C22×D4—21no—
147C22×D4—21no—
148C22×D4C22×C412yes—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4C2×D412yes—
154C23×C4C2×D412yes—
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2411yescharacteristic
158C22×C8—22no—
159C22×C8—22no—
160C4⋊Q8—22no—
161C4⋊Q8—22no—
162C4⋊Q8—48no—
163#127—22no—
164#127—22no—
165#124—48no—
166#124—42no—
167#124—42no—
168#123—48no—
169#123—48no—
170C22⋊Q8—416no—
171#121—22no—
172#121—22no—
173#121—22no—
174#121—22no—
175#121—42no—
176#121—42no—
177C4×Q8—24no—
178C4×Q8—24no—
179C4×Q8—24no—
180C4×Q8—24no—
181C4×Q8—24no—
182C4×Q8—24no—
183C4×Q8—24no—
184C4×Q8—24no—
185C4×Q8—432no—
186C4×D4—24no—
187C4×D4—24no—
188C4×D4—42no—
189C4×D4—42no—
190#117—42no—
191#117—42no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—48no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4C22⋊C412yes—
211C2×C4⋊C4C22⋊C412yes—
212C2×C4⋊C4C22⋊C414yes—
213C2×C4⋊C4C22⋊C414yes—
214C2×C4⋊C4C22×C412yes—
215C2×C4⋊C4C22×C414yes—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—28no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—28no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—28no—
226C2×C42—22no—
227C2×C42—24no—
228C2×C42—44no—
229C2×C42—44no—
230C2×C42—42no—
231C2×C42—42no—
232C2×C42C22×C414yes—
233C2×C42C22×C411yescharacteristic
234C2×C42C22×C411yescharacteristic
235C2×C42C2×D414yes—
236C2×C42C2×D412yes—
237C2×C42C2×D412yes—
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D411yescharacteristic
240C2×C42#4112yes—
241C2×C42#4112yes—
242C4⋊C8—42no—
243C4⋊C8—42no—
244C22⋊C8—22no—
245C22⋊C8—22no—
246C22⋊C8—22no—
247C22⋊C8—22no—
248C4×C8—42no—
249C4×C8—42no—
250#95—28no—
251#95—22no—
252#95—28no—
253#95—24no—
254#95—21no—
255#95—21no—
256#95—22no—
257#95—22no—
258#95C2×D412yes—
259#95C2×D414yes—
260#95C2×D412yes—
261C24—22no—
262C24C22×D411yescharacteristic
263#41—84no—
264#41—84no—
265#41—84no—
266#41—84no—
267C2×Q8—24no—
268C2×Q8—24no—
269C2×Q8—24no—
270C2×Q8—24no—
271C2×Q8—28no—
272C2×Q8—28no—
273C2×Q8—28no—
274C2×Q8—28no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8#10218yes—
279C2×Q8#10218yes—
280C2×D4—22no—
281C2×D4—22no—
282C2×D4—44no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—82no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—24no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—42no—
307C22×C4—44no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—42no—
313C22×C4—44no—
314C22×C4—42no—
315C22×C4—44no—
316C22×C4—42no—
317C22×C4—41no—
318C22×C4—42no—
319C22×C4—41no—
320C22×C4—44no—
321C22×C4—44no—
322C22×C4—44no—
323C22×C4C2×C22⋊C414yes—
324C22×C4C2×C22⋊C411yescharacteristic
325C22×C4C2×C22⋊C411yescharacteristic
326C22×C4C2×C22⋊C414yes—
327C22×C4C2×C22⋊C411yescharacteristic
328C22×C4C2×C22⋊C411yescharacteristic
329C22×C4#12012yes—
330C22×C4#12012yes—
331C22×C4#12312yes—
332C22×C4#12312yes—
333C22×C4C2×D814yes—
334C22×C4C2×D812yes—
335C22×C4C2×SD1614yes—
336C22×C4C2×SD1612yes—
337C22×C4AGL(1,ℤ/8)12yes—
338C22×C4#13712yes—
339C22×C4C23×C411yescharacteristic
340C22×C4C22×D411yescharacteristic
341C22×C4C22×D411yescharacteristic
342C22×C4C22×D411yescharacteristic
343C22×C4C2×#4111yescharacteristic
344C22×C4C2×#4111yescharacteristic
345C2×C8—44no—
346C2×C8—44no—
347C2×C8—42no—
348C2×C8—42no—
349C4⋊C4—28no—
350C4⋊C4—28no—
351C4⋊C4—24no—
352C4⋊C4—24no—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C4⋊C4—44no—
356C4⋊C4—48no—
357C4⋊C4—416no—
358C4⋊C4—416no—
359C4⋊C4—42no—
360C4⋊C4—44no—
361C4⋊C4—416no—
362C4⋊C4—48no—
363C4⋊C4—82no—
364C22⋊C4—42no—
365C22⋊C4—42no—
366C22⋊C4—42no—
367C22⋊C4—42no—
368C22⋊C4—44no—
369C22⋊C4—416no—
370C22⋊C4—88no—
371C42—24no—
372C42—24no—
373C42—24no—
374C42—24no—
375C42—416no—
376C42—44no—
377C42—48no—
378C42C4×D412yes—
379C42C4×D412yes—
380C42C4×D412yes—
381C42C4×D412yes—
382C42#12112yes—
383C42#12112yes—
384C42#12112yes—
385C42#12112yes—
386C23—22no—
387C23—22no—
388C23—44no—
389C23—42no—
390C23—42no—
391C23—44no—
392C23—42no—
393C23C22×C22⋊C411yescharacteristic
394C23C2×#12011yescharacteristic
395C23C2×#12311yescharacteristic
396C23C22×D811yescharacteristic
397C23C22×SD1611yescharacteristic
398C23C2×AGL(1,ℤ/8)11yescharacteristic
399C23C2×#13711yescharacteristic
400Q8—216no—
401Q8—216no—
402Q8—84no—
403D4—82no—
404D4—82no—
405D4—84no—
406D4—82no—
407D4—82no—
408D4—82no—
409D4—82no—
410D4—84no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—24no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—28no—
423C2×C4—24no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—42no—
434C2×C4—44no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—44no—
438C2×C4—42no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—84no—
449C2×C4—82no—
450C2×C4—84no—
451C2×C4—82no—
452C2×C4—88no—
453C2×C4#38612yes—
454C2×C4#38612yes—
455C2×C4#38612yes—
456C2×C4#38612yes—
457C2×C4C2×#10218yes—
458C2×C4C2×#10212yes—
459C2×C4C2×#10212yes—
460C2×C4#41712yes—
461C2×C4#41712yes—
462C2×C4#44714yes—
463C2×C4#44814yes—
464C2×C4#48014yes—
465C2×C4#48114yes—
466C2×C4C2×C4×D411yescharacteristic
467C2×C4C2×C4×D411yescharacteristic
468C2×C4C2×#12111yescharacteristic
469C2×C4C2×#12111yescharacteristic
470C8—82no—
471C8—82no—
472C22—22no—
473C22—22no—
474C22—22no—
475C22—22no—
476C22—21no—
477C22—21no—
478C22—21no—
479C22—21no—
480C22—82no—
481C22—84no—
482C22—84no—
483C22—82no—
484C22—82no—
485C22C2×#38611yescharacteristic
486C22#288412yes—
487C22C22×#10211yescharacteristic
488C22C2×#41711yescharacteristic
489C22#290112yes—
490C22C2×#44711yescharacteristic
491C22C2×#44811yescharacteristic
492C22C2×#48011yescharacteristic
493C22C2×#48111yescharacteristic
494C4—28no—
495C4—22no—
496C4—22no—
497C4—21no—
498C4—21no—
499C4—21no—
500C4—21no—
501C4—44no—
502C4—44no—
503C4—84no—
504C4—82no—
505C4—82no—
506C4—84no—
507C4#187414yes—
508C4#187414yes—
509C2—21no—
510C2—21no—
511C2—21no—
512C2—21no—
513C2—162no—
514C2#1761812yes—
515C2#1762812yes—
516C2C2×#187411yescharacteristic
517C2#2098011yescharacteristic
518C2#2098611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#2351C2semidirect
3. #21818C2semidirect
6. #9156C2semidirect
7. #9148C2semidirect
8. #8026C2semidirect
9. #8014C2semidirect
18. #2548C4semidirect
19. #2546C4semidirect
21. #2538C4semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#351782
C2×#23512
#218182
C2×#15172
C2×#15022
#91562
#91482
#80262
#80142